Maxima Basics #4

Maxima Version 5.20.1
wxMaxima Version 0.8.4

(%i1) load(draw)$

(%i2) load("C:/Documents and Settings/Compaq_Owner/My Documents/Maxima/vect1.mac")$

(1)
2D Parametric curves defined by vector-valued functions...

(%i3) /* Position vector */ r: [2*cos(t)-cos(2*t),2*sin(t)-sin(2*t)];

(%i4) /* Velocity vector */ v: diff( r, t );

(%i5) /* Acceleration vector */ a: diff( r, t, 2 );

Sketch the graphs of position, velocity, and acceleration.

(%i6) wxdraw2d( nticks = 100,
        color = blue, parametric( r[1], r[2], t, 0, 2*%pi ),
        color = green, parametric( v[1], v[2], t, 0, 2*%pi),
        color = red, parametric( a[1], a[2], t, 0, 2*%pi )
 )$

Add some details to the graph...

(%i7) wxdraw2d( xaxis = true, xaxis_width = 2, xaxis_color = gray, xlabel = "x",
        yaxis = true, yaxis_width = 2, yaxis_color = gray, ylabel = "y",
        nticks = 100,
        color = blue, parametric( r[1], r[2], t, 0, 2*%pi ),
        color = green, parametric( v[1], v[2], t, 0, 2*%pi),
        color = red, parametric( a[1], a[2], t, 0, 2*%pi )
 )$

(2)
continuing with the same position vector as above...

(%i8) /* Unit tangent vector */ T: normalize( v );

Sketch the graph of r(t) along with the tangent vector at t=2.5.

(%i9) /* r(t) at t=2.5 */ rpt: at( r, t=2.5 );

(%i10) /* T(t) at t=2.5 */ Tpt: at( T, t=2.5 );

(%i11) wxdraw2d( xrange = [-3,3], yrange = [-3,3],
        xaxis = true, xaxis_color = gray,
        yaxis = true, yaxis_color = gray,
        nticks = 100,
        parametric( r[1], r[2], t, 0, 2*%pi ),
        color = red, point_type = 7, points( [ rpt ] ),
        color = blue, head_length = 0.1, vector( rpt, Tpt )
 )$

(3)
Continuing with the same position vector as above, add the principal unit
normal vector to the graph.

(%i12) /* Compute T'(t) */ Tp: diff(T,t);

(%i13) /* Principal unit normal at t=2.5 */ Npt: normalize( at( Tp, t=2.5 ) );

Npt doesn't look orthogonal to Tpt because of the aspect ratio of the viewing window.

(%i14) wxdraw2d( xrange = [-3,3], yrange = [-3,3],
        xaxis = true, xaxis_color = gray,
        yaxis = true, yaxis_color = gray,
        nticks = 100,
        parametric( r[1], r[2], t, 0, 2*%pi ),
        color = red, point_type = 7, points( [ rpt ] ),
        color = blue, head_length = 0.1, vector( rpt, Tpt ),
        color = green, vector( rpt, Npt )
 )$

(4)
Continuing with the same position vector as above, compute the curvature at t=2.5.

(%i15) /* Compute |T'(2.5)| */ temp1: norm( at( Tp, t=2.5 ) );

(%i16) /* Compute |v(2.5)| */ temp2: norm( at( v, t=2.5 ) );

(%i17) /* Curvature at t=2.5 */ Kpt: temp1/temp2;

(5)
Continuing with the same position vector as above, sketch the circle of
curvature at t=2.5

(%i18) radius: 1/Kpt;

(%i19) center: rpt + radius*Npt;

(%i20) wxdraw2d( xrange = [-3,3], yrange = [-3,3],
        xaxis = true, xaxis_color = gray,
        yaxis = true, yaxis_color = gray,
        nticks = 100,
        parametric( r[1], r[2], t, 0, 2*%pi ),
        color = red, point_type = 7, points( [ rpt ] ),
        color = blue, head_length = 0.1, vector( rpt, Tpt ),
        color = green, vector( rpt, Npt ),
        color = orange, points( [ center ] ),
        transparent = true, ellipse( center[1], center[2], radius, radius, 0, 360 )
 )$